The compactness of a weakly singular integral operator on weighted Sobolev spaces. (English) Zbl 1323.46026

Summary: It is shown that the weakly singular integral operator \(\int_{-1}^{1}\big(\phi(\tau)/|\tau -t|^{\gamma}\big)\,d\tau\), where \(0< \gamma< 1\), maps the weighted Sobolev space \(W_{p;\alpha,\beta}^{(n)}(\Omega)\) compactly into itself for \(1< p< \infty\), \(0< \alpha+1/q, \beta+1/q< 1\) and \(n\in \mathbb{N}_0\).


46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)


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